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Energies 2015, 8, 2337-2357; doi:10.3390/en8042337
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energies
ISSN 1996-1073
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Article
Sintered Nickel Powder Wicks for Flat Vertical Heat Pipes
Geir Hansen *, Erling Næss † and Kolbeinn Kristjansson †
Department of Energy and Process Engineering, Norwegian University of Science and Technology,
Kolbjørn Hejes vei 1a, 7491 Trondheim, Norway; E-Mails: [email protected] (E.N.);
[email protected] (K.K.)
†
These authors contributed equally to this work.
* Author to whom correspondence should be addressed; E-Mail: [email protected];
Tel.: +47-7359-3938; Fax: +47-7359-3580.
Academic Editor: Enrico Sciubba
Received: 12 January 2015 / Accepted: 16 March 2015 / Published: 25 March 2015
Abstract: The fabrication and performance of wicks for flat heat pipe applications produced
by sintering a filamentary nickel powder has been investigated. Tape casting was used as an
intermediate step in the wick production process. Thermogravimetric analysis was used to
study the burn-off of the organic binder used and to study the oxidation and reduction
processes of the nickel. The wicks produced were flat, rectangular and intended for liquid
transport in the upwards vertical direction. Rate-of-rise experiments using heptane were used
to test the flow characteristics of the wicks. The wick porosities were measured using
isopropanol. The heat transfer limitation constituted by the vapour static pressure and the
capillary pressure was discussed. The influence on wick performance by using pore former
in the manufacturing was studied. When Pcap/Psat > 1, the use of a pore former to increase the
wick permeability will always improve the wick performance. When Pcap/Psat < 1, it was
shown that if the effective pore radius and the permeability increase with an equal percentage
the overall influence on the wick capacity is negative. A criterion for a successful pore
former introduction is proposed and the concept of a pore former evaluation plot is presented.
Keywords: wick; sintering; pore former; rate-of-rise; heat pipe
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1. Introduction
The concept of heat pipes was first described in 1942, but the real development of heat pipes started
in the 1960s [1]. Properly designed heat pipes can transport heat at very high rates with a small
temperature difference between the cold and the warm end. This is possible because the heat transport
inside the heat pipe is based on evaporation, transport and condensation of a suitable working fluid, and
thereby also obtaining an apparent thermal conductivity several hundred times that of solid materials
such as copper [2]. The heat pipe interior contains a wick, in which the condensate returns from
the condenser section to the evaporator section. The driving force in the condensate transport process is
the capillary pressure created by the wick and working fluid interaction, in some cases the capillary
pressure is supported by the gravity force. The gravity force contributes to the transport process when
the condenser section is located above the evaporator section, i.e., the heat pipe is gravity-assisted [3].
The heat pipe is gravity-opposed when the condenser section is located below the evaporator.
Wicks can be produced by several different techniques and from different materials, such as woven
metal wires, metal foams, sintered metal powders, or they can be made as grooves in the wall of the heat
pipe [4]. Bimodal wicks are made by adding metal powders to metal foams or felts [5–7]. In some cases
the wick capacity can be improved by creating internal arteries inside it, forming low-friction flow paths
for the liquid [4]. De Schampeleire et al. [3] stated that “the number of possible wick types is virtually
endless”. They investigated wicks for water/copper heat pipes having 6 mm outer diameter and 200 mm
length, for cooling of electronic devices. Specifically, a novel wick made of metal fibers was compared
to more traditional wicks made of a screen mesh and a sintered powder, respectively. Under gravity-assisted
conditions the screen mesh wick performed best, because of its superior permeability. High permeability
means low frictional pressure loss for the fluid flowing through the wick. In the gravity-opposed
orientation the screen mesh wick and the metal fibre wick both outperformed the sintered powder wick [3].
However, the properties of sintered powder wicks can be modified and improved by use of pore formers.
The pore former, usually an organic material, is mixed with the metal powder and thereafter “burnt off”,
leaving a void behind. The burn-off process using an air or oxygen atmosphere is usually a separate
process prior to the sintering. The pore former will normally increase the pore size and thereby also
increase the effective pore radius of the wick. Different heat pipe applications and different working
conditions require different wick properties, it is therefore important to know how the different
production parameters such as pore formers affect the wick properties. Li et al. [8] investigated
experimentally how much the forming pressure and the use of a microcrystalline cellulose (MCC) pore
former affected the wick porosity and capillary pumping performance of a sintered powder wick for loop
heat pipes, and developed a correlation between porosity, forming pressure and MCC addition. The use
of a pore former for wick capacity improvement is analysed further in the current study. A criterion for
successful pore former introduction is derived, and the novel concept of a pore former evaluation plot is
presented. The ideal pore former can be totally removed from the wick structure, for instance by heat
treatment (burn-off) and/or by chemical treatment. The suitability of pure graphite powder and carbon
fibers as pore formers are considered in the current study and the required burn-off temperature for
graphite powder is determined experimentally. Information about the required burn-off temperature is
essential when pore former material is to be selected, if the burn-off temperature is too high the wick
material may be damaged.
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Compatibility with the working fluid is often the main wick material selection criterion, and the wick
material is often a metal. Incompatibility can lead to generation of non-condensable gases, which can
displace the working fluid and thereby hamper the heat transfer [9]. However, material compatibility is
not sufficient. Proper thermal and/or chemical cleaning of the wick is equally important in order to avoid
corrosion, non-condensable gases and to ensure proper wetting of the wick by the working fluid [10,11].
The importance of the wick in the cleaning process is emphasized by the fact that the total surface area
of the wick is much larger than the surface area of the container; the surface of the wick can therefore
contain much more adsorbed unwanted components than the container wall.
Wicks of sintered nickel powder have been used for many years in heat pipes [1,4]. Nickel has relatively
high thermal conductivity, which is an advantage when heat is to be conducted across the wick as in
traditional heat pipes. For loop heat pipes high thermal conductivity of the wick is a disadvantage [8,12].
Nickel is compatible with alkali metals, frequently used as working fluids for high temperature heat pipe
applications [1]. In the current study a wick of sintered nickel powder was developed, and its suitability
for gravity-opposed operation with potassium as working fluid is evaluated. Specifically, the intended
application was a special type of heat pipe, having a flat wick in the evaporator section only. Figure 1
shows the working principle.
Figure 1. A flat wick attached to a solid substrate subject to a uniform heat flux causing
liquid evaporation from the free wick surface. Liquid (condensate) is pumped by the wick
from a reservoir at the bottom of the wick. The experiments utilized to characterize the wick
in this study were performed without the heat addition and without the solid substrate.
The filamentary nickel powder Inco Type 255 is a popular raw material for wick production [5,8,13].
The filamentary particles make it possible to produce wicks with porosities far above the porosity of any
wick produced of monodisperse spherical particles. The porosity of a wick produced from monodisperse
spherical particles is theoretically between 26% and 47%, while the porosity of a wick of filamentary
particles can be up to ca. 90% [13]. The wick developed in the current study was made of sintered Inco
Type 255 nickel powder. The following parameters are known to affect the properties of sintered Inco Type
255 nickel powder [14]:
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Sintering time.
Sintering temperature.
Sintering atmosphere.
Type and amount of pore former (spacing agent).
Pressure from weights applied during sintering (forming pressure).
Long sintering time, high sintering temperature, and high forming pressure will in general lead to
compact wick structures, having low porosity, low permeability and a small effective pore radius.
The minimum sintering time, temperature and forming pressure providing sufficient mechanical strength
should therefore be the best choices in order to maintain an open, high porosity wick structure.
The sintering atmosphere is very important when nickel powder is used, because of the susceptibility of
nickel to oxidise at high temperatures. The surface properties of nickel oxide will be different from that
of pure nickel, and this will in general affect the interfacial tensions and the contact angle between the
solid and the working fluid. In order to avoid oxidation of the nickel, the sintering of nickel powder can
be carried out in protective/reducing gas atmosphere [12] or in vacuum [7]. Similarly, nickel oxides,
e.g., formed during burn-off, can be reduced to pure nickel by heat treatment under a reducing
atmosphere such as H2 (5 vol%) and N2 (95 vol%) [14]. The mass increase and colour change from grey
to green indicate nickel oxide formation, and vice versa for the reduction. Thermogravimetric analysis
(TGA) was utilized in the currrent work to study the oxidation of the nickel powder, burn-off, and to
study reduction of nickel oxide in a 5% H2 + 95% N2 atmosphere. Such data were needed in the
preparation of the burn-off, sintering and oxide reduction programs.
The wick structure has to be robust to be able to withstand handling and thermal strains and stresses,
especially important for high temperature heat pipes and when the heat load is unilateral. In the current
study a robust wick was obtained utilizing tape casting [15] as an intermediate step in the wick
production. The required burn-off time and temperature level for the removal of the additives used in
the tape casting were determined by thermogravimetric analysis, and are reported.
Good thermal contact between the wick and the heat pipe wall reduces the overall heat transfer
resistance from the outside of the evaporator to the working fluid. The thermal contact resistance
between the wick material and the evaporator surface is eliminated when the wick is sintered to the
evaporator surface. Burn-off and sintering directly on the evaporator surface was investigated
experimentally in this study.
Wicks for heat pipe use are usually characterized by their porosity, permeability, and effective pore
radius related to a specific fluid. A standardized method for porous samples can be applied for the
porosity measurements [16]. Permeability can be measured in forced flow experiments [17], but has also
been determined together with the effective pore radius in spontanous free flow experiments,
i.e., rate-of-rise experiments [13,18]. The rate-of-rise experiment was chosen in the current study.
Model fluids (e.g., acetone, heptane, methanol) are often utilized in rate-of-rise experiments [13],
especially when the real working fluid is hazardous (e.g., alkali metals). Heptane was chosen in the
current study. Permeability is independent of the type of fluid, but the effective pore radius is not.
In order to convert the effective pore radius from one fluid to another, the contact angles of the respective
fluids on the wick powder surface are required. The contact angle of the model fluid (heptane) on porous
nickel was derived from capillary rise experiments, as described by Hiemenz [19]. According to
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Hiemenz [19], the capillary method for contact angle determination “is not highly reliable, but preferable
compared to any technique based on the exterior surface of the plug as a liquid support”. The “plug” is
in this case the wick. Alkali metals are known to have “strong wetting characteristics” against metals [20].
Hence, the contact angle of potassium on nickel was assumed to be zero [21].
The heat transfer capacity of heat pipes can be restricted by many different heat transfer limitations,
which have been given considerable attention in the literature [4,22]. The current paper presents and
analyzes a heat transfer limitation which has not been given attention previously; the static pressure
limitation. It is shown theoretically how the static pressure limitation will limit the performance of the
designed nickel powder wick with potassium as working fluid. To summarize, the following topics are
investigated in this study:





Method of fabrication of the wick.
Selection of burn-off and sintering temperature programs from thermogravimetric analysis.
Determination of the wick characteristics.
Prediction of the wick maximum heat transfer capacity with potassium.
Criterion for pore former selection.
2. Method of Fabrication of the Wick
Wicks were made by sintering pure Inco Type 255 nickel filament. During this study many wicks
were produced by different methods (tape casting or mould, with or without pore former). A production
method based on tape casting of a slip without pore former was found to be the most successful with
regards to the mechanical properties of the wick. In the following this approach is described in detail,
as well as the testing and results for a wick produced by this method. Some results from the work with
pore former have also been included. Moulding was attempted, but sufficient mechanical strength was
not achieved, hence this method was considered inferior.
In the tape casting, a modified batching procedure for “High-surface-area oxide powder, oxidizing
sintering procedure” [15] was used, summarized as: Sinter powder (Inco 255), distilled water and
dispersant (Darvan C) were mixed by rolling for about 12 h. A binder (polyvinyl alcohol), a plasticizer
(polyethylene glycol) and a defoamer (polypropylene glycol) were then added to the mixture, and the
mixture was mixed by rolling for an additional 12 h, after which it was ready for tape casting using a
tape casting machine. The tape casting produced a thin (0.2–0.3 mm thick) tape of the mixture. The tape
was let to dry for about 12 h in ambient air atmosphere, and was next cut into pieces of desired size.
Due to shrinkage during subsequent burn-off and sintering the pieces had to be cut 20%–30% larger than
the desired final wick size. The desired wick thickness was achieved by stacking a number of tape pieces
on top of each other. For the wick presented in this paper 16 layers of tape were required in order to have
a total thickness of 4 mm before burn-off and sintering.
In order to achieve good contact between the tapes and sufficient mechanical strength of the wick,
it was necessary to apply pressure upon the pile of tapes during burn-off and sintering. Alumina weights
which provided a pressure of 1,300 Pa were used. Ideally, the wick should be sintered directly onto the
evaporator surface of the heat pipe as this would eliminate thermal contact resistance between the wick
and the evaporator surface. Burn-off and sintering directly on the evaporator surface was tested,
but failed because of incomplete burn-off, probably caused by insufficient gas diffusion in the pile of
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tapes which were covered by the heat pipe walls and the alumina weights. It was therefore decided to
produce the wick outside of the heat pipe, and thereafter attach it to the evaporator surface by a suitable
method. Following this new approach, the pile of tapes was put on an alumina plate, and the weights put
on top of the pile. This was then all put in the oven which had continous synthetic air flushing for
effective burn-off of the organic material. The burn-off and sintering temperature program was
developed from information from thermogravimetric analysis (TGA), together with known sintering
temperature requirements for a mechanical strong wick.
3. Selection of Burn-off and Sintering Temperature Programs from Thermogravimetric Analysis




Thermogravimetric analysis of pure Inco 255 nickel powder in air was carried out to study
the oxidation.
Thermogravimetric analysis was carried out for the tape cast tape in order to determine the
programs for burn-off and reduction. In this test a 900 µL crucible was filled with
approximately 40 layers of tape. The burn-off TGA was carried out using synthetic air
atmosphere, and the reduction in a H2 (5%) and N2 (95%) atmosphere.
A thermogravimetric analysis with pure graphite powder in air was performed. Pore former
was not used in the fabrication of the final wick presented in this study, but some work was
performed to analyse the suitability of the graphite powder as pore former. Graphite was
selected because it leaves very little residues after burn-off. The graphite powder (from
Sigma-Aldrich) had particle sizes smaller than 20 µm.
A thermogravimetric analysis was also performed for the burn-off of a tape casted tape
consisting of approximately equal amounts of Inco 255 nickel powder and graphite powder,
plus the tape casting additives.
The results from these four TGA experiments are presented graphically in Section 5—Results
and Discussion.
4. Determination of the Wick Characteristics
4.1. Capillarity and Flow Characteristics
When a wick is used for liquid transport in the vertical upward direction as shown in Figure 1, its capacity
is highly dependent on the effective pore radius and the permeability of the wick structure. The driving
force is the capillary pressure, which is a function of the liquid wetting, the liquid surface tension and the
pore radius of the wick. The wetting of a solid by a liquid is often expressed by the contact angle. The
effective pore radius includes the contact angle information and is defined from a capillary tube model [13]:
Pcap 
2 cos( ) 2

r
reff
(1)
The relation between the effective radii of two fluids having different contact angles is:
reff , fluid 1
reff , fluid 2

cos  fluid 2
cos  fluid 1
(2)
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In the wick the primary forces opposing the capillary pressure are the hydrostatic pressure and the
frictional pressure drop of the flow. Of less importance are fluid momentum changes and friction
between the liquid and the substrate, both disregarded in the present analysis. The hydrostatic pressure
of the liquid column can be expressed as:
Pg    l  v  gH   l gH
(3)
where H is the height of the liquid level in the wick relative to the fluid bath. The pressure drop due to
friction is calculated from Darcy’s law [13]:
dPf

dy
V 


m 
 Ac 
(4)
The highest mass flow rate is at the liquid inlet end (i.e., bottom) of the wick. For a constant evaporation
rate per unit height of wick, the mass flow rate in the wick becomes a linear function of position (y):
m  y   m y 0 
m y 0 y
H
y

 m y 0 1  
 H
(5)
The total frictional pressure drop up to height H is determined by integration of Equation (4) between
the limits y = 0 and y = H; with Equation (5):
 m y 0 H
m 
dy  
 Ac 
2 Ac 
0
H
Pf   
(6)
The overall fluid force balance is comprised of the capillary pressure (Equation (1)), the static head
(Equation (3)) and the fluid friction (Equation (6)) and can be expressed as:
 m y  0 H
2
  gH 
0
2 Ac 
reff
(7)
The wick liquid pumping capacity ( m y 0 ) is found by solving Equation (7) provided that the wick
properties (reff and к) are known. The resulting total evaporative heat transfer capacity of the wick is then:

2 Ac   2
Q  m y 0 h fg 
  gH  h fg


H   reff

(8)
Here, the wick properties reff (effective pore radius) and к (permeability) are of particular interest.
In general, these parameters yield opposite influences; a smaller effective pore radius will give a higher
capillary pressure but will also tend to reduce the permeability. Wick optimization is therefore a task of
finding a wick that provides the required liquid lifting height (H) and simultaneously provides sufficient
mass transport capacity ( m y 0 ).
4.2. Wick Porosity
The dimensions of the wick were after the sintering carefully adjusted by use of a high speed cutting
tool. The cutoff pieces were used in the porosity experiment, based on the international standard ISO
5017 [16], using isopropanol.
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4.3. Rate-of-Rise
The wick flow characteristics were tested in a rate-of-rise experiment, as described by Holley and
Faghri [13]. Heptane was used as working fluid, and was introduced into the wick from a Petri dish
placed on a balance while the wick was hanging from a stand. Figure 2 shows the setup.
Figure 2. The rate-of-rise setup.
The mass uptake was measured by a balance (AT261 Deltarange with 10 µg resolution, manufactured
by Mettler Toledo, Greifensee, Switzerland) connected to a PC running the LabView™ logging
program. The evaporation rate from the surface of the heptane pool of the Petri dish was measured by
recording the weight loss vs. time. Next, the wick tip was brought in contact with the heptane, and the
weight loss was measured by the balance during the capillary rise of heptane in the wick. When the wick
tip was submerged, the liquid meniscus around the wick imposed an upward force on the Petri dish
causing a small drop in the force measured on the balance. Contrary to this, a small force increase on the
balance caused by the submerged part of the wick is expected. These forces partly cancel each other and
they do not change much during the experiment, i.e., they are not significant for the shape of the
mass-time curve used for the data reduction. No corrections of the recorded curves were therefore made.
Finally, the combined evaporation rate from the wick and the Petri dish was measured when the liquid
had reached the maximum height in the wick. The evaporation rate from the wick was determined by
subtracting the evaporation rate of the pool in the Petri dish from the combined evaporation rate.
Similarly, the weight measurements during capillary rise were corrected for the evaporation from the
heptane pool surface of the Petri dish. A mathematical model for the rate-of-rise experiment taking
evaporation from the wick into account was developed by Holley and Faghri [13], whose solution is
presented as Equation (9). The derivation of Equation (9) is involved, however, it describes the
rate-of-rise of a liquid which is spontaneously wetting an initially dry vertically mounted wick and takes
evaporation into account. A high evaporation rate decreases the rate-of-rise. Equation (9) is discussed in
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more detail in the paper by Holley and Faghri [13]. A MatLab™ script was used in the present work to
fit the measured mass-time history to Equation (9):
  C1  C2 


y
y 
C1 ln 1    C2 ln 1    
t

2
C
C
A
c
1
2 


(9)
The constants C1 and C2 were the roots to the quadratic equation, Equation (10), where C is the
variable [13]:
C2 
2  2 gAc
4 Ac
0
C

reff 
(10)
The MatLab™ script determined the best pair of (reff, к) by comparing the closed form solutions
with experimental data and selecting the pair of (reff, к) which caused the minimum mean absolute
deviation (MAD).
5. Results and Discussion
5.1. Thermogravimetric Analysis and Temperature Programs
5.1.1. Nickel Oxide Formation
Nickel oxide formation occured from around 300 °C, as can be seen in Figure 3, showing the
mass/temperature plot of a thermogravimetric analysis performed using pure Inco 255 nickel powder
in air. Increasing recorded mass indicates oxide formation on the powder surface.
Figure 3. Thermogravimetric analysis of pure Inco 255 nickel powder in air.
5.1.2. Burn-off and Reduction
The result of a thermogravimetric analysis with the actual tape casted tape is shown in Figure 4.
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Figure 4. Thermogravimetric analysis of tape cast tape burn-off and reduction.
The first 24 h of this TGA temperature program was the burn-off sequence, using synthetic air
atmosphere. The sample weight started to decrease rapidly at about 140 °C, indicating burn-off of the
organic material in the tape. At about 350 °C the sample weight started increasing, likely due to a very
high nickel oxide formation rate. The temperature was maintained at 600 °C for one hour. The sample
weight then stabilized, indicating that the high rate of oxide formation had ended. After 24 h the
temperature was at room temperature again and the gas flow was changed from synthetic air to a mixture
of 5% (vol) H2 + 95% (vol) N2. Heating was started again, and at a temperature of about 350 °C the
sample weight started decreasing, indicating that the nickel oxide was being reduced to pure nickel.
Above 350 °C the rate of weight loss during reduction was almost constant, even when the temperature
was increased up to 600 °C. This indicated that another parameter than temperature was limiting the rate
of reduction, most likely the gas diffusion in the sample (pile of tapes) in the crucible, and/or limited
molecular transport on/in the surface of the nickel oxide. The efficiency of the burn-off and reduction
process could not be measured/quantified, hence the surface was investigated using microscope.
No signs of unburned organics or green nickel oxide were found.
5.1.3. The Temperature Program
The final oven temperature program chosen for the burn-off and sintering process of the real wick is
shown in Figure 5. The TGA-data of Figure 4 were utilized in the planning of the process.
The requirements were:



Sufficient time and temperature for the burn-off.
Sufficient time and temperature for proper sintering and acceptable mechanical properties.
Sufficient time and temperature for the reduction of the nickel oxide.
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In general, too much time and/or too high temperature should be avoided in all parts of the program
as this would make the wick unnecessary compact.
The first stage, i.e., the burn-off in synthetic air, was carried out at a maximum level of about 360 °C.
The gas composition was changed from synthetic air to a mixture of 5%H2 + 95%N2 after about 29 h.
The second stage, i.e., the sintering and reduction, started with a short temperature peak of about 580 °C
to ensure proper sintering and mechanical properties of the wick. In order avoid too much compaction
of the wick the temperature was then reduced to 400 °C. To ensure sufficient time for the reduction,
it was decided to keep the wick for 8 h at about 400 °C.
Figure 5. The temperature in the oven during the burn-off, the sintering and the reduction.
The tape layer structure could be observed after the sintering, but the resulting wick was
mechanically strong.
5.1.4. Pure Graphite Powder
The thermogravimetric analysis with the pure graphite powder in air is shown in Figure 6.
The result shows that a burn-off temperature of 800 °C will oxidize all the graphite to CO or CO2 gas.
5.1.5. Tape Containing Pore Former
The thermogravimetric analysis performed for the burn-off in air of a tape casted tape consisting of
approximately equal amounts of Inco 255 nickel powder and graphite powder, plus the tape casting
additives is shown in Figure 7.
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Figure 6. Thermogravimetric analysis with pure graphite powder in air, heating rate 5 K/min.
Figure 7. Thermogravimetric analysis of tape casted tape in air, heating rate 5 K/min.
The weight changes of the sample can be explained as follows: From 150 °C to 400 °C the weight
decreased because the tape casting additives were being burnt off. From around 300 °C nickel oxide was
formed, causing a weight increase. From around 600 °C the graphite started oxidizing and leaving the
sample. Just above 800 °C the weight stabilized, indicating that all the graphite was gone and no more
nickel oxide was being formed.
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5.2. The Contact Angle
From a series of capillary rise experiments with nickel foam wicks and heptane the contact angle of
heptane on nickel was determined to 58.4° [23]. For potassium on nickel a contact angle of 0° was
assumed [21].
5.3. Wick Porosity
The porosity measured from three samples of the wick produced by tape casting were ε1 = 0.801,
ε2 = 0.79 and ε3 = 0.806, which yielded an average result of ε ≈ 0.80. The applied measurement method
ISO 5017 declares that “the values of porosity shall be given to the nearest 0.1% (V/V)” [16]. In the
current study the sizes of the samples were smaller than specified by the standard, this could have lead
to somewhat larger variations in the results than expected from the standard. The uncertainty is set equal
to the measurement variability, i.e., ±0.7%.
5.4. Rate-of-Rise
The height vs time data from the rate-of-rise experiment for the wick produced by tape casting is
shown in Figure 8. The height values were calculated numerically from the mass data from the balance,
taking the evaporation into account. The evaporation rate from the wick was determined from the
experiment. The measured points in Figure 8 are completely covering the “Fit” line generated from the
rate-of-rise theory.
Figure 8. Rate-of-rise data for the wick produced by tape casting.
The permeability was determined to 0.108·10−12 m2 and the effective pore radius with heptane to
0.826·10−6 m. From Equation (2) the effective pore radius with potassium then became 0.43 µm. The
sensitivity of these values to changes in key variables is discussed in Section 5.6—Sensitivity Analysis.
The thermophysical data for heptane and potassium were taken from Faghri [4].
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5.5. Prediction of the Wick Maximum Heat Transfer Capacity with Potassium
Static Pressure Limitation
The capacity of the wick made by tape casting, 2.825 mm thick, operated with potassium at 500 °C
would according to Equation (8) have been approximately 94 kW/m2 uniform heat flux over the entire
surface of 25 × 150 mm. Unfortunately the actual capacity for potassium at 500 °C is much less
(only 844 W/m2), due to a heat transfer limitation which has not been given attention in literature: In a
heat pipe the capillary pressure can never be higher than the vapour static pressure of the working fluid.
For alkali metals, which have very low vapour static pressures, this limitation is of particular interest,
especially where large driving pressures are needed, like in long vertical heat pipes with the capillary
force working against the gravity. This limitation is termed the “static pressure limit” as it is the vapour
static pressure of the working fluid that limits the available driving pressure.
A reduction of the wick height reduces the pressure losses and increases the heat transfer capacity.
The tape casted wick could have handled a uniform heat flux of 9069 W/m2 if its height was reduced to
50 mm. An example of how the static pressure limit of the wick appears in a temperature vs. heat flux
plot is shown in Figure 9. The plot is for the tape casted wick, 150 mm and 50 mm high, with potassium
as working fluid. The capillary pressure is equal to the vapour static pressure at the peaks where the lines
representing SPL and CPL intercept. The vapour static pressure of potassium is the limiting pressure for
the operation temperatures below the interception point. For temperatures above the interception point
the capillary pressure becomes the limiting pressure. For the tape casted wick and using potassium,
the interception is obtained at a temperature of about 883 °C. However, this temperature is so high that
it would cause sintering of this wick during use.
Figure 9. The theoretical performance of the tape cast wick with potassium working fluid.
SPL = Static pressure limited. CPL = Capillary pressure limited.
The exponential form of the SPL curves is caused by the increase in vapour static pressure of the
potassium with increasing temperature. The drop in capacity in the CPL region with increasing T is due to
Energies 2015, 8
2351
the combined effect of the fluid physical properties (viscosity, surface tension, density and latent heat
of vapourization).
5.6. Sensitivity Analysis
A sensitivity analysis was carried out to see the influence of changes in temperature, porosity, evaporation
rate and length measurements on the heat transfer capacity of the wick produced by tape casting.
The parameters were varied one by one and the corresponding combinations of effective pore radius and
permeability determined. The corresponding heat transfer capacities were calculated by use of Equation (8).
The Influence of Parameter Variations Above and Below Pcap/Psat = 1
It was noticed that small changes in the parameters could cause quite large changes in the effective
pore radius and permeability. This finding was also reported by Nam et al. [24] who, in order to
overcome the large uncertainties, predicted the permeability by numerical simulations, and used this
permeability value together with the rate-of-rise data to determine the effective pore radius. Separate
determination of the permeability was also used by Deng et al. [12,25].
Despite of the high sensitivity of (reff, к) to changes in the parameters, the corresponding heat transfer
capacities were almost unaffected as long as the capillary pressure was the limiting pressure, indicating
that the new sets of effective pore radius and permeability were all on an iso-capacity line. However,
the large changes in the permeability propagated to large uncertainties in the calculated capacities for
temperatures where the vapour static pressure of the potassium was the limiting pressure, since in this
range the large permeability changes were not balanced by corresponding changes in the capillary pressure.
Changes in the heat transfer capacity were calculated for the temperatures 850 °C and 1050 °C to
illustrate how changes in the parameters had different impacts above and below Pcap/Psat = 1. The results
are presented in Table 1.
Table 1. The heat transfer capacity sensitivity due to parameter changes above and below Pcap/Psat = 1.
Parameter
Change
Temperature
Porosity
Evaporation rate
Cross sectional area
+1.2 K
+1%
+1%
+1%
Change in heat transfer
capacity at 850 °C
(vapour static pressure limited)
−28%
−40%
−28%
−47%
Change in heat transfer
capacity at 1050 °C
(capillary pressure limited)
−0.2%
−0.9%
−0.05%
−6%
The large differences between the sensitivities for the calculated capacities above and below
Pcap/Psat = 1 are clearly seen. Plots of the mean absolute deviation (MAD) between the calculated and
measured time-height-data did show that there were many pairs of (reff, к) that had near identical MAD.
All these different pairs of (reff, к) would give an apparently close to perfect curve fit of the height-time
measurements (Figure 8). The calculated sets of (reff, к) for each parameter perturbation in Table 1 varied
significantly (as implicitly indicated by the vapour-static-pressure-limited performance sensitivity),
but had no significant impact on the capillary-pressure-limited performance. Hence, the rate-of-rise
results are useful in the CPL range, even though the individual real values of reff and к are “unknown”.
Energies 2015, 8
2352
The permeability has to be known with higher certainty for more reliable wick capacity predictions in
the temperature range where the vapour static pressure is limiting.
5.7. Criterion for Pore Former Selection
Improved wick performance for potassium/alkali metals would be obtained in both the vapour static
pressure limited region and the capillary pressure limited region if the permeability of the wick could be
increased without affecting the effective pore radius. However, an increase in the permeability is
normally accompanied by an increase in the effective pore radius, and it is therefore necessary to
consider the consequences of changes in these parameters together. For wicks made of sintered powder
the permeability (and thereby also inevitably the effective pore radius) can be increased by use of a pore
former. The permeability and the effective pore radius will generally not experience the same relative
change. The consequence for the wick capacity in the vapor static pressure limited region depends on
the permeability change, and the consequence in the capillary pressure limited region depends on the
relative changes of the permeability and the effective pore radius. The consequence for the wick
performance will generally be different in the vapour static pressure limited region and the capillary
pressure limited region, because the vapour static pressure is unaffected by changes in the effective pore
radius while the capillary pressure is directly a function of the effective pore radius.
In the vapour static pressure limited region the wick performance will always be improved by
increased permeability. In order for the pore former to cause general capacity improvement in the
capillary pressure limited region, the relative changes in effective pore radius and permeability have to
satisfy the criterion for wick capacity improvement by use of pore former, which is derived below.
From Equation (8) it is seen that a proportional increase of the effective pore radius and the permeability
actually reduces the wick capacity. Wick improvement requires that the permeability increases more
than the effective pore radius, and the exact requirement can be derived from Equation (8). If the increase
in effective pore radius is expressed by the factor kr and the increase in permeability is expressed by the
factor kк then Equation (8) becomes:

2 k Ac   2
Q  m y 0 h fg 
  gH  h fg


H   reff kr

(11)
By combining Equation (11) and Equation (8) the relation between kк and kr appears as:
2
  gH
reff
k 
2
  gH
reff kr
(12)
In Figure 10 the factor kк is plotted against the factor kr for different effective pore radii using
Equation (12), with H = 0.1 m and properties for heptane at temperature 20 °C.
Energies 2015, 8
2353
Figure 10. Pore former evaluation plot for H = 0.1 m and heptane at 20 °C.
The lines in Figure 10 are iso-capacity lines, i.e., the wick capacities are unchanged along the lines.
For effective pore radii up to about 15 µm the lines are almost linear, i.e., the capillary term is completely
dominating. With increasing effective pore radii the lines become more curved, i.e., in order to maintain
the wick capacity the permeability has to increase more than the effective radius. When an iso-capacity
line curves upwards the overall effect of a change kr = kк on the wick capacity is negative. The use of a pore
former is only successful if it leads to a point above the line, see Figure 11. For instance, in Figure 10,
for reff = 35 µm a 20% increase of the effective pore radius has to be accompanied by more than 60%
increase of the permeability in order to have an overall heat transfer capacity improvement for the wick.
Figure 11. Pore former evaluation plot.
One way to achieve larger increase in the permeability than in the effective pore radius may be to use
a pore former consisting of long filaments, creating longitudinal channels in the sintered metal powder.
A potential candidate is using carbon fibers arranged in the wick flow direction.
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2354
6. Conclusions







The capillary pressure inside a heat pipe can never exceed the vapour static pressure of the
working fluid. This will limit the heat transfer capacity and the useful operating temperature
range for heat pipes with wicks with small effective pore radii operated with working fluids
with low vapour static pressures, for instance alkali metals.
Determination of the effective pore radius and the permeability from a rate-of-rise experiment
alone can lead to great uncertainties about the actual values. Many different pairs of (reff, к)
give good curve fits, and in such case they can all be used for approximate wick capacity
predictions for (Pcap/Psat < 1). For wick capacity predictions for (Pcap/Psat > 1) the permeability
has to be known with higher accuracy.
In the vapour static pressure limited region the use of a pore former which increases the wick
permeability will always improve the wick performance.
In the capillary pressure limited region, a pore former evaluation plot may be made in order to
analyse the effects of using a pore former. If the effective radius and permeability increase with
an equal percentage the overall effect on the wick heat transfer capacity is negative. A pore
former which increases the permeability more than the effective pore radius would be a better
choice than a powderous pore former. One such candidate can be carbon fibers.
The oxidation of Inco 255 nickel powder accellerates at about 350 °C.
The reduction of nickel oxide in 5%H2 + 95%N2 accellerates at about 350 °C.
In order to burn off a graphite powder pore former a temperature of about 800 °C is required.
Acknowledgments
The authors would like to thank Elkem and the Research Council of Norway for funding this work.
Thanks to Mr. Rhodri Bowen of Vale Inco Europe Ltd for supplying Inco 255 nickel powder. Thanks to
Dr. Vegar Øygarden for assistance on pure nickel powder TGA and to Dr. Ove Paulsen for valuable and
inspiring talks on sintering.
Author Contributions
Geir Hansen carried out the experimental work, data reduction, and prepared the first draft of the
paper. The MatLab™-code for the rate-of-rise analysis and the pore former success criterion were made
by Kolbeinn Kristjansson. Geir Hansen and Erling Næss have equally contributed on the analysis and
paper preparation.
Symbols
Ac
C
E
g
H
hfg
wick cross sectional area, m2
constant
Deviation
gravity constant, g = 9.81 m/s2
height, m
latent heat of evaporation, kJ/kg
Energies 2015, 8
kк
kr
Q
factor expressing change in permeability
factor expressing change in radius
mass flow rate, kg/s
pressure, Pa
heat flow rate, W
r
reff
t
V
V
y
radius, m
effective radius, m
time, s
volume
velocity, m/s
height, m

m
P
2355
Greek symbols
Г
ε
θ
к
µ
ρ
σ
evaporation rate per unit height of wetted wick, kg/(m·s)
porosity
solid-liquid-vapour contact angle, degrees
permeability, m2
dynamic viscosity, kg/(m·s)
density, kg/m3
surface tension, N/m
Subscripts
cap
eff
f
g
l
sat
v
capillary
effective (pore radius)
friction
gravity
liquid
saturated
vapour
Conflicts of Interest
The authors declare no conflict of interest.
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